Asymptotic behaviour of nonlocal p-Laplacian reaction-diffusion problems

Abstract

In this paper, we focus on studying the existence of attractors in the phase spaces L2(Ω) and Lp(Ω) (among others) for time-dependent p-Laplacian equations with nonlocal diffusion and nonlinearities of reaction-diffusion type. Firstly, we prove the existence of weak solutions making use of a change of variable which allows us to get rid of the nonlocal operator in the diffusion term. Thereupon, the regularising effect of the equation is shown applying an argument of a posteriori regularity, since under the assumptions made we cannot guarantee the uniqueness of weak solutions. In addition, this argument allows to ensure the existence of an absorbing family in W1,p 0 (Ω). This leads to the existence of the minimal pullback attractors in L2(Ω), Lp(Ω) and some other spaces as Lp∗−(Ω). Relationships between these families are also established.